Collaborative Research: Nonparametric Theory on Manifolds of Shapes and Images, with Applications to Biology, Medical Imaging and Machine Vision.

这个合作项目的大部分焦点是基于地标的形状的分析，其中，为了识别、辨别或诊断的目的，通常在专家的帮助下，在2-D或3-D中观察到k-ad，即物体或场景上的一组k个点或地标。根据数据收集或记录的方式，物体的适当形状是由G组变换下的轨道空间指定的最大不变量。特别地，k-ads的Kendall形状空间在标度运动和欧氏刚体运动下是不变的。虽然这是生物学和医学成像中许多问题的合适选择，但其他形状概念，如仿射形状和投影形状，在机器视觉和生物信息学中也很重要。所有这些空间都是可微流形，通常具有测量长度和角度的自然黎曼结构。基于黎曼结构的统计分析被认为是内在的。在其他情况下，通过将流形M等变嵌入到向量空间E中来寻找适当的距离。相应的统计分析称为非本征分析。寻找合适的黎曼结构和等变嵌入是本项目的目标之一，这对于所提出的统计推断是至关重要的。为Fre‘chet均值的存在建立广泛的条件，作为Fre’chet函数的唯一极小值，离Q分布随机形状的期望平方距离对于统计推断是重要的；它是本项目追求的目标，特别是对于内在分析，因为它从形状理论开始就一直是一个突出的开放问题。从飞机上拍摄的两张(或多张)航拍照片重建场景是仿射形状分析中的研究问题之一。本文提出的投影形状分析的潜在应用包括人脸识别和机器人-机器人视觉识别场景。几何对象或流形数据的统计分析是一个令人兴奋和具有挑战性的研究领域，其中统计理论和微分几何密不可分，实现需要创新的算法和高速的计算。这里提出的项目涉及在这种情况下非参数方法的发展，它还必须解决相关的几何问题和实施问题。私人投资促进机构过去在这方面取得的进展，为目前的项目奠定了基础。所提出的统计分析具有广泛的应用，特别是在生物学和生物信息学、健康科学和机器视觉方面。根据这一项目，私人投资机构计划在理论和实际实施方面培训本科生和研究生，以及至少一名博士后研究员。这进一步延续了私人投资机构目前在这方面的活动。此外，还在网站上提供了计算算法和代码，以创建和传播这项研究及其应用。

Much of the focus of this collaborative project is on the analysis of landmark based shapes in which a k-ad, i.e., a set of k points or landmarks on an object or a scene are observed in 2-D or 3-D, usually with expert help, for purposes of identification, discrimination, or diagnostics. Depending on the way the data are collected or recorded, the appropriate shape of an object is the maximal invariant specified by the space of orbits under a group G of transformations. In particular, Kendall's shape spaces of k-ads are invariant under scaling and Euclidean rigid motions. While this is a proper choice for many problems in biology and medical imaging, other notions of shape such as affine shape and projective shape are important in machine vision and bioinformatics. All these spaces are differentiable manifolds, often with natural Riemannian structures for measuring lengths and angles. The statistical analysis based on Riemannian structures is said to be intrinsic. In other cases, proper distances are sought via an equivariant embedding of the manifold M in a vector space E. Corresponding statistical analysis is called extrinsic. Finding proper Riemannian structures and equivariant embeddings is one of the objectives of this project, which is crucial for the statistical inference proposed. Establishing broad conditions for the existence of the Fre´chet mean, as the unique minimizer of the Fre´chet function the expected squared distance from a Q-distributed random shape is important for statistical inference; and it is a goal of the project to pursue, especially for intrinsic analysis where it has remained an outstanding open problem from the inception of shape theory. Reconstruction of a scene from two (ormore) aerial photographs taken from a plane is one of the research problems in affine shape analysis. Potential applications of projective shape analysis proposed here include face recognition and robotics-for robots to visually recognize a scene.Statistical analysis of data on geometric objects, or manifolds, is an exciting and challenging field of research, where statistical theory and differential geometry are inextricably intertwined, and implementation requires innovative algorithms and high speed computation. The project proposed here deals with the development of nonparametric methodology in this context, which must also resolve associated geometric issues and problems of implementation. Past progress in this field by the PIs has laid the foundation for the present project. The statistical analysis proposed has wide ranging applications, especially in biology and bioinformatics, health sciences, and machine vision. Under this project, the PIs plan to train both undergraduate and graduate students, as well as at least one postdoctoral fellow, in theory and in its practical implementation. This continues much further the present activities of the PIs in this regard. In addition, computational algorithms and codes are made available on websites to create and disseminate this research and its applications.